What is 21.63846384638463846... as a Fraction?

21.63846 equals 721213333 as a fraction.

How do you turn 21.63846 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.63846 repeating into a fraction, begin writing this simple equation:

n = 21.63846 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (3846) and 1 digit in the non-repeating part (6). Multiply both sides by 10⁵ = 100000.

100000 × n = 2163846.3846 (equation 2)

Step 3: Now subtract equation 1 from equation 2 to cancel the repeating block (or repetend) out.

100000 × n = 2163846.3846
    10 × n = 21.63846
 99990 × n = 2163630

216363099990 could be the answer, but it still can be put in the simplest form, i.e., reduced.

To simplify this fraction, divide the numerator and denominator by 30 (the GCF - greatest common factor).

n = 216363099990 = 2163630 ÷ 3099990 ÷ 30 = 721213333. So,

21.63846 = 721213333 as the lowest possible fraction.

As the numerator is greater than the denominator, we have an improper fraction, so we can also express it as a mixed number, thus 721213333 is also equal to 2121283333 when expressed as a mixed number.

The repeating decimal 21.63846 (vinculum notation) has a repeated block length of 4. It is also represented as 21.6384638463846... (ellipsis notation) which equals approximately 21.638463846384638463846 (decimal approximation)(*).

The recurring decimal 21.63846 can be written as a ratio of two integers having 72121 as the numerator and 3333 as the denominator. So, it is a rational number (named after ratio). It can be shown that a number is rational if its decimal representation is repeating or terminating.

(*) At present, there is no single universally accepted notation or phrasing for repeating decimals.